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Unformatted text preview: ; (1.3.11a) where hj = xj ; xj
Thus, U (x) is constant on xj ; j = 1 2 : : : N: 1 (1.3.11b) xj ] and is given by the rst divided di erence
U (x) = cj ; cj 1
x 2 xj 1 xj ]:
Substituting (1.3.11) and a similar expression for V (x) into (1.3.9b) yields
S (V U ) =
p dj 1 dj ] ;1=hj ;1=hj 1=hj ] cjc 1 dx
0 1 ; ; 0 ; 0 j ; ; j or AS (V U ) = dj 1 dj ]
; Z xj
p ;1=h2 ;1=hj2 dx cjc 1 :
2 2 ; The integrand is constant and can be evaluated to yield p
Kj = h AS (V U ) = dj 1 dj ]Kj cjc 1
; ; j 1 ;1 ;1 1: (1.3.12) The 2 2 matrix Kj is called the element sti ness matrix. It depends on j through hj ,
but would also have such dependence if p varied with x. The key observation is that
Kj can be evaluated without knowing cj 1, cj , dj 1, or dj and this greatly simpli es the
automation of the nite element method.
The evaluation of AM proceeds similarly by substituting (1.3.10) into (1.3.9d) to
M (V U ) =
q dj 1 dj ] j 1 j 1 j ] cjc 1 dx:
; ; j ; xj;1 j ; ; ; j With q a constant, the integrand is a quadratic polynomial in x that ma...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14